Autonomously navigating robots have long been a tough challenge facing engineers. The recent push to develop micro-aerial vehicles for practical military, civilian, and industrial use has added a significant power and time constraint to the challenge. In contrast, animals, from insects to humans, have been navigating successfully for millennia using a wide range of variants of the ultra-low-power computational system known as the brain. For this reason, we look to biological systems to inspire a solution suitable for autonomously navigating micro-aerial vehicles. In this dissertation, the focus is on studying the neurobiological structures involved in mammalian spatial navigation. The mammalian brain areas widely believed to contribute directly to navigation tasks are the Head Direction Cells, Grid Cells and Place Cells found in the post-subiculum, the medial entorhinal cortex, and the hippocampus, respectively. In addition to studying the neurobiological structures involved in navigation, we investigate various neural models that seek to explain the operation of these structures and adapt them to neuromorphic VLSI circuits and systems. We choose the neuromorphic approach for our systems because we are interested in understanding the interaction between the real-time, physical implementation of the algorithms and the real-world problem (robot and environment). By utilizing both analog and asynchronous digital circuits to mimic similar computations in neural systems, we envision very low power VLSI implementations suitable for providing practical solutions for spatial navigation in micro-aerial vehicles.